Theorem alrimii 30451

Description: A lemma for introducing a universal quantifier, in inference form. (Contributed by Giovanni Mascellani, 30-May-2019.)

Hypotheses

Ref	Expression
alrimii.1	|- F/ y ph
alrimii.2	|- ( ph -> ps )
alrimii.3	|- ( [. y / x ]. ch <-> ps )
alrimii.4	|- F/ y ch

Assertion

Ref	Expression
alrimii	|- ( ph -> A. x ch )

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)   ps( x, y)   ch( x, y)

Proof of Theorem alrimii

Step	Hyp	Ref	Expression
1		alrimii.1	. . 3 |- F/ y ph
2		alrimii.2	. . . 4 |- ( ph -> ps )
3		alrimii.3	. . . 4 |- ( [. y / x ]. ch <-> ps )
4	2, 3	sylibr 212	. . 3 |- ( ph -> [. y / x ]. ch )
5	1, 4	alrimi 1825	. 2 |- ( ph -> A. y [. y / x ]. ch )
6		nfsbc1v 3356	. . 3 |- F/ x [. y / x ]. ch
7		alrimii.4	. . 3 |- F/ y ch
8		sbceq2a 3348	. . 3 |- ( y = x -> ( [. y / x ]. ch <-> ch ) )
9	6, 7, 8	cbval 1994	. 2 |- ( A. y [. y / x ]. ch <-> A. x ch )
10	5, 9	sylib 196	1 |- ( ph -> A. x ch )

Syntax hints:   -> wi 4   <-> wb 184   A.wal 1377   F/wnf 1599   [.wsbc 3336

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-sbc 3337

This theorem is referenced by: (None)